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Dehn-Nielsen-Baer theorem

The Dehn-Nielsen-Baer theorem states, for closed orientable surfaces $\Sigma^2_g$ of positive genus, $g \geq 1$, that the canonical homomorphism from their extended mapping class group $MCG^{\pm}$ (including orientation-reversing maps) to the outer automorphism group $Out(-)$ of their fundamental group $\pi_1(-)$ (at any basepoint, using that surfaces are assumed to be connected) is an isomorphism:

$$MCG^{\pm}(\Sigma^2_g) \xrightarrow{\phantom{-}\sim\phantom{-}} Out\big( \pi_1(\Sigma^2_g) \big) \,.$$

References

The first published argument, credited to Max Dehn (whose proof has been lost, cf. Stillwell 1987):

• Jakob Nielsen: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. I-III, Acta Math. 50 1 (1927) 189-358 [MR1555256, doi:10.1007/BF02547775], Acta Math. 53 1 (1929) 1-76 [MR1555290, doi:10.1007/BF02547566], Acta Math. 58 1 (1932) 87-167 [MR1555345, doi:10.1007/BF02547775]

Detailed review of the original arguments by Max Dehn and Nielsen, as far as recoverable:

• John Stillwell: The Dehn-Nielsen Theorem, Appendix in: Max Dehn: Papers on Group Theory and Topology Springer (1987)363-395 [doi:10.1007/978-1-4612-4668-8_17, pdf]

Textbook accounts:

• Benson Farb, Dan Margalit, chapter 8 of: A primer on mapping class groups, Princeton Mathematical Series, Princeton University Press (2012) [ISBN:9780691147949, jstor:j.ctt7rkjw]

Generalization to orbifolds:

• Hansjörg Geiges, Jesús Gonzalo: Generalised spin structures on 2-dimensional orbifolds, Osaka J. Math. 49 (2012) 449–470